Is there a text-caret Unicode character?

Well I think it’s safe to say that no, there is no such character. Part of the proposal process for requesting a Unicode glyph is to perform a search, which I did, but it came up empty.

In addition, every other search I made came up empty, including one with ShapeCatcher which found several decent stand-ins (below), but none that works quite right (is full-height).

I guess until they add one, we’ll have to use CSS to style and size one of these to fit the part.

ⵊꕯ𝖨𐌆𝙸Ꮖ

There is a Unicode character for caret, and it's called "caret". It is U+2038: http://www.fileformat.info/info/unicode/char/2038/index.htm

You can find it in the General Punctuation category in some character mapping accessories (such as Gnome Character Map). It is not widely available, but it can be found in at least in 50 fonts that are listed on http://www.fileformat.info/info/unicode/char/2038/fontsupport.htm. You might not be happy with the design of the character in these fonts, but that's an entirely different issue.

If you're simply looking for a picture, and it doesn't need to be one character, or even span one line, you could try drawing it in the style of ASCii art:

▝▚▞▘
   ▌
▗▞▚▖

Since, in rich text documents, Unicode can carry any font size, you could even potentially make the cursor/caret the same size as adjacent text ... just be sure to keep it in a mono-space font face.

This example makes use of:

U+259D  ▝   Quadrant upper right

U+259A  ▚   Quadrant upper left and lower right

U+259E  ▞   Quadrant upper right and lower left

U+2598  ▘   Quadrant upper left

U+258C  ▌   Left half block

U+2597  ▗   Quadrant lower right

U+259E  ▞   Quadrant upper right and lower left (again)

U+259A  ▚   Quadrant upper left and lower right (again)

U+2596  ▖   Quadrant lower left

For more information about block elements: https://en.wikipedia.org/wiki/Block_Elements